Example 1
\(2x + 3 > 7 \rightarrow x > 2\)
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Theory & Formula
A linear inequality is similar to a linear equation, but instead of equality, it uses inequality symbols (<, >, ≤, ≥).
Key rules for solving inequalities:
- Addition/Subtraction: Adding or subtracting the same value on both sides preserves the inequality
- Multiplication/Division (positive): Multiplying or dividing by a positive number preserves the inequality
- Multiplication/Division (negative): Multiplying or dividing by a negative number reverses the inequality
- Solution set: An inequality typically has infinitely many solutions, represented on a number line or in interval notation
\(ax + b < c \rightarrow x < \frac{c - b}{a}\) (if \(a > 0\))
Worked Examples
Example 2
\(-3x + 5 \leq 14 \rightarrow x \geq -3\)
Example 3
\(5x - 2 < 13 \rightarrow x < 3\)
Interval Notation
\(x \geq 2\) is written as \([2, \infty)\)